4 CURVE OF 



Let D n, D'n', I)"n'\ be the consecutive 

 values of the ordinate. 



Let Aw, A n\ A n'\ be the consecutive 

 values of the string and vi^eight n. 



Draw n r, nir', perpendicular to D^w' and 

 D" rl! : and draw vl' tt, perpendicular to D' w' pro- 

 duced ; and draw np, nfp^ perpendicular to An\ 



and A 7i\ 



Let n L n'^ be two elements of another curve, 

 infinitely near the first curve, nnfn"; and draw 

 L ff perpendicular to Ari\ cutting A n^ in s. Then, 

 because the value of a quantity, near its minimum, 

 does not differ from the value of that quantity, 

 at its minimum, the time through w L + time 

 through L n!^ = time through w ^^' + time through 

 n^n'^ : and, therefore t ,nn' -- t ,nlj — t.lwn!' — 

 t.n'n". (a) 



Then, substituting in the mechanical theorem 



